Discrete Mathematics 2010-2022

Primitive lattice in $Z^n$ are a generalization of the concept of primitive vectors: a rank $d$ subgroup of $Z^n$ is called primitive if there is no other subgroup of the same rank that properly contains it. In two papers from 1998 and from 2015, Schmidt proved a counting statement for primitive lattices of any rank $1<d<n$, taking into account their shapes (similarity classes modulo rotation and rescaling, namely projections into $SO(d) \backslash SL_d(R)/ SL_d(Z))$, and directions (the subspaces that they span, namely projections into the Grasmannian $GR(d,n)$). We extend upon this counting statement, and also consider the shapes of the orthogonal complements of these lattices. Moreover, we introduce the concept of flags of primitive lattices, and extend this counting statement to them as well.

This is joint work with Yakov Karasik.

Meeting number: 121 139 9332

Password: fvNiUjUP332

Link:

https://tugraz.webex.com/tugraz/j.php?MTID=mb910f1333b42f13ca5aca85391713524

I'll discuss our recent proof of a conjecture of Talagrand, a fractional version of the ``expectation-threshold'' conjecture of Kahn and Kalai. As a consequence of this result, we resolve various (heretofore) difficult problems in probabilistic combinatorics and statistical physics. Time permitting, I’ll also outline how these methods can be used to determine the threshold of the square of the Hamilton cycle, a stubborn hitherto open problem in random graph theory.

Meeting link:
\[
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Meeting number:
137 149 1265

Password:
JYc3B3dunG2

Horospherical products of two Gromov hyperbolic spaces unify the construction of metric spaces such as the Diestel-Leader graphs, the SOL geometry or the treebolic space. These examples, which are coming either from geometric group theory or from the study of solvable Lie groups, share similar rigidity properties.
In this talk we will first recall all the bases required to construct the horospherical products. Then we will study the large scale geometry of a family of them through a description of their geodesics and visual boundary. 

webex meeting, Thursday, 21 Jan 2021, 10:30 (Rome, Stockholm, Vienna)

Meeting number: 121 802 9686 ‚ Password: u3Mpx5sJF2H

https://tugraz.webex.com/tugraz/j.php?MTID=macdcb248c84d8d6069425a0db76a2fc6

meeting to be opened at 10:30, the talk will start at 11:00